Problem:

There is a 4 x 6 rectangle, a circle with radius 1, and a circle with radius 3. The two circles are externally tangent to each other and two adjacent vertices are on the centers of the circles. If the area that is inside the rectangle but outside of the circles can be expressed as , where b and c are relatively prime positive integers, what is a + b + c?

Solution:

let x be the area of inside the rectangle but outside of the circles (orange color from the figure))

Area of rectangle = Area of quarter circle of radius 1 + Area of quarter circle of radius 3 + x

As the problem states that:

The area that is inside the rectangle but outside of the circles can be expressed as , where b and c are relatively prime positive integers.

We understand that 5 is a relatively prime positive integer and 2 is also a relatively positive prime integer

the variable a can be written as which is equal to 7.63943726841

where x can be expressed as

where a = 7.63943726841

b = 5 (relatively prime positive integer)

c = 2 (relatively prime positive integer)

both 2 and 5 are relatively prime positive integer or coprimes.

simplifiying the equation

Solving for the area that is inside the rectangle but outside of the circles (orange color) from the figure gives:

x = 5.13943726841 π

x = 16.146 sq. uni ts

What is a + b + c?

a + b + c = 7.63943726841 + 5 + 2

a + b + c = 14.6394372684

Answer:

14.6394372684

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